For minimization problems involving vectors of objective functions, a Pareto optimal point is a solution whose associated vector of objective function values cannot be uniformly improved upon by any other point. Finding Pareto optimal solutions is particularly difficult when the functions are not continuous and have local minima, and when the computation of function values is time-consuming or costly.
The problem of determining the set of all Pareto optimal points traditionally has been important for many scientific disciplines, for example, economics, finance, telecommunication, and electrical engineering. In recent years, several new application fields have emerged. For example, the detection of Pareto optimal regions has become increasingly important in aeronautical as well as structural mechanical design, and in computational fluid dynamics; see for example (Thévenin and Janiga, 2008). In that setting, genetic algorithms are typically used for solving multi-objective optimization problems. Extensive computational tests have established that these methods perform well, but also require a large number of function evaluations. If the function evaluation subproblem is computationally expensive, applicability of the algorithms is limited.
Problem Definition: Let X be an axis-parallel rectangle, for short rectangle, of n-dimensional Euclidean space En; that is, for some n-dimensional vectors l and u satisfying l≦u , X={x|l≦x≦u}. Define ƒ(x) to be an m-dimensional vector where the kth entry is a function ƒk(x) from X to the line E1. Now find the Pareto optimal set X* for the minimization of ƒ(x) over X, that is,X*={x* ε X|∀x ε X[ƒ(x)≦ƒ(x*)ƒ(x)=ƒ(x*)]}If m=1, then ƒ(x) is a scalar, and X* is the set of vectors x* minimizing the function over X.
There are numerous prior approaches and methods for solving the stated problem under various assumptions. For an overview, see (Coello Coello et al, 1996). For many real-world problem instances, the function ƒ(x) may not be continuous and may have local minima. In such cases, deterministic techniques tend to be ineffective, and stochastic optimization methods such as Simulated Annealing (Kirkpatrick et al, 1983), Monte Carlo methods (Robert and Casella, 2004), Tabu Search (Glover and Laguna, 1997), and Evolutionary Computation (Bäck, 1996; Goldberg, 1989) seem more appropriate.
As a result of the foregoing, there is a need for an alternative approach where fewer function evaluations are needed.